Physlib.QFT.PerturbationTheory.WickAlgebra.WicksTheoremNormal

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators associated with a field specification $\mathcal{F}$. The time-ordered product of these operators is given by the sum over all Wick contractions $\Lambda$ that involve only equal-time pairings: $$\mathcal{T}(\phi_0 \phi_1 \dots \phi_{n-1}) = \sum_{\Lambda \in \text{EqTimeOnly}(\phi_s)} \epsilon(\Lambda, \phi_s) \cdot \text{timeContract}(\Lambda) \cdot \mathcal{T}(:\! [\phi_s]_\Lambda^{uc} \!:)$$ where: - $\mathcal{T}$ denotes the time-ordering operator. - $\text{EqTimeOnly}(\phi_s)$ is the set of Wick contractions $\Lambda$ such that for every contracted pair $\{i, j\} \in \Lambda$, the operators $\phi_i$ and $\phi_j$ satisfy the time-ordering relation in both directions (physically, they occur at the same time). - $\epsilon(\Lambda, \phi_s)$ is the sign factor ($\pm 1$) resulting from the permutations of fermionic operators. - $\text{timeContract}(\Lambda)$ is the central element of the Wick algebra representing the product of pairwise contractions for all pairs in $\Lambda$. - $:\dots:$ (or $\mathcal{N}$) denotes the normal-ordering operator. - $[\phi_s]_\Lambda^{uc}$ is the sub-list of field operators from $\phi_s$ that are not contracted by $\Lambda$.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators associated with a field specification $\mathcal{F}$. The time-ordered product of these operators is equal to the time-ordered product of their normal-ordered form plus a sum over all non-empty Wick contractions that involve only equal-time pairings: $$\mathcal{T}(\phi_0 \phi_1 \dots \phi_{n-1}) = \mathcal{T}(:\! \phi_0 \phi_1 \dots \phi_{n-1} \!:) + \sum_{\substack{\Lambda \in \text{EqTimeOnly}(\phi_s) \\ \Lambda \neq \emptyset}} \epsilon(\Lambda, \phi_s) \cdot \text{timeContract}(\Lambda) \cdot \mathcal{T}(:\! [\phi_s]_\Lambda^{uc} \!:)$$ where: - $\mathcal{T}$ denotes the time-ordering operator. - $:\dots:$ (or $\mathcal{N}$) denotes the normal-ordering operator. - $\text{EqTimeOnly}(\phi_s)$ is the set of Wick contractions $\Lambda$ such that for every contracted pair $\{i, j\} \in \Lambda$, the operators $\phi_i$ and $\phi_j$ satisfy the time-ordering relation in both directions (physically, they occur at the same time). - $\epsilon(\Lambda, \phi_s)$ is the sign factor ($\pm 1$) resulting from permutations of fermionic operators. - $\text{timeContract}(\Lambda)$ is the central element of the Wick algebra representing the product of pairwise contractions for all pairs in $\Lambda$. - $[\phi_s]_\Lambda^{uc}$ is the sub-list of field operators from $\phi_s$ that are not contracted by $\Lambda$.

Let $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators associated with a field specification $\mathcal{F}$. The time-ordered product of the normal-ordered operators is given by the time-ordered product of the operators themselves minus the sum over all non-empty Wick contractions $\Lambda$ consisting only of equal-time pairings: $$\mathcal{T}(\mathcal{N}(\phi_0 \dots \phi_{n-1})) = \mathcal{T}(\phi_0 \dots \phi_{n-1}) - \sum_{\substack{\Lambda \in \text{EqTimeOnly}(\phi_s) \\ \Lambda \neq \emptyset}} \epsilon(\Lambda, \phi_s) \cdot \text{timeContract}(\Lambda) \cdot \mathcal{T}(\mathcal{N}([\phi_s]_\Lambda^{uc}))$$ where: - $\mathcal{T}$ denotes the time-ordering operator. - $\mathcal{N}$ denotes the normal-ordering operator. - $\text{EqTimeOnly}(\phi_s)$ is the set of Wick contractions $\Lambda$ such that for every contracted pair $\{i, j\} \in \Lambda$, the operators $\phi_i$ and $\phi_j$ occur at the same time (i.e., they satisfy the time-ordering relation in both directions). - $\epsilon(\Lambda, \phi_s)$ is the sign factor ($\pm 1$) resulting from the permutation of fermionic operators. - $\text{timeContract}(\Lambda)$ is the product of the pairwise contractions for all pairs in $\Lambda$. - $[\phi_s]_\Lambda^{uc}$ is the sub-list of field operators from $\phi_s$ that are not contracted by $\Lambda$.

Let $\mathcal{F}$ be a field specification, $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators, $\mathcal{T}$ be the time-ordering operator, and $\mathcal{N}$ be the normal-ordering operator. For any Wick contraction $\Lambda$, let $\text{sign}(\Lambda)$ be its statistical sign factor, $\text{timeContract}(\Lambda)$ be the product of its pairwise time-ordered contractions, and $[\Lambda]^{uc}$ be the list of uncontracted operators. We define the Wick term as $\text{wickTerm}(\Lambda) = \text{sign}(\Lambda) \cdot \text{timeContract}(\Lambda) \cdot \mathcal{N}([\Lambda]^{uc})$. The time-ordering of the product of fields in $\phi_s$ is given by: $$\mathcal{T}(\phi_s) = \sum_{\substack{\Lambda \in \text{Wick}(\phi_s) \\ \neg \text{HaveEqTime}(\Lambda)}} \text{sign}(\Lambda) \cdot \text{timeContract}(\Lambda) \cdot \mathcal{N}([\Lambda]^{uc}) + \sum_{\substack{\Lambda \in \text{Wick}(\phi_s), \Lambda \neq \emptyset \\ \text{EqTimeOnly}(\Lambda)}} \text{sign}(\Lambda) \cdot \text{timeContract}(\Lambda) \left( \sum_{\substack{\Lambda' \in \text{Wick}([\Lambda]^{uc}) \\ \neg \text{HaveEqTime}(\Lambda')}} \text{wickTerm}(\Lambda') \right)$$ where: - $\text{HaveEqTime}(\Lambda)$ is the property that $\Lambda$ contains at least one pair of operators evaluated at the same time. - $\text{EqTimeOnly}(\Lambda)$ is the property that all pairs in $\Lambda$ are evaluated at the same time. - The first sum is over contractions $\Lambda$ with no equal-time pairs. - The second term is a nested sum where the outer sum is over non-empty contractions $\Lambda$ consisting solely of equal-time pairs, and the inner sum is over contractions $\Lambda'$ of the remaining fields that contain no equal-time pairs.

Let $\mathcal{F}$ be a field specification and $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators. Let $\mathcal{T}$ denote the time-ordering operator and $\mathcal{N}$ denote the normal-ordering operator. For any Wick contraction $\Lambda$ on the indices of $\phi_s$, let $\epsilon(\Lambda, \phi_s)$ be the statistical sign factor, $\text{timeContract}(\Lambda)$ be the product of pairwise contractions, and $[\phi_s]_\Lambda^{uc}$ be the sub-list of uncontracted operators. Define the Wick term as $\text{wickTerm}(\Lambda) = \epsilon(\Lambda, \phi_s) \cdot \text{timeContract}(\Lambda) \cdot \mathcal{N}([\phi_s]_\Lambda^{uc})$. The time-ordered product of the normal-ordered operators satisfies the following inductive identity: $$\mathcal{T}(\mathcal{N}(\phi_0 \dots \phi_{n-1})) = \sum_{\substack{\Lambda \\ \neg \text{HaveEqTime}(\Lambda)}} \text{wickTerm}(\Lambda) + \sum_{\substack{\Lambda \neq \emptyset \\ \text{EqTimeOnly}(\Lambda)}} \epsilon(\Lambda, \phi_s) \cdot \text{timeContract}(\Lambda) \left( \sum_{\substack{\Lambda' \\ \neg \text{HaveEqTime}(\Lambda')}} \text{wickTerm}(\Lambda') - \mathcal{T}(\mathcal{N}([\phi_s]_\Lambda^{uc})) \right)$$ where: - $\text{HaveEqTime}(\Lambda)$ is the property that $\Lambda$ contains at least one pair of operators evaluated at the same time. - $\text{EqTimeOnly}(\Lambda)$ is the property that all pairs in $\Lambda$ are evaluated at the same time. - The first sum is over all Wick contractions $\Lambda$ of $\phi_s$ that contain no equal-time pairs. - The second sum is over all non-empty Wick contractions $\Lambda$ of $\phi_s$ consisting solely of equal-time pairs. - Inside the second sum, $\Lambda'$ denotes Wick contractions of the uncontracted list $[\phi_s]_\Lambda^{uc}$.

For a given field specification $\mathcal{F}$, let $\mathcal{T}$ denote the time-ordering operator and $\mathcal{N}$ denote the normal-ordering operator. For an empty list of field operators (representing the identity element $1$ in the field operator algebra), the time ordering of its normal ordering is equal to the sum of the Wick terms over all Wick contractions $\Lambda$ of length $0$ that do not contain equal-time pairs: $$\mathcal{T}(\mathcal{N}(1)) = \sum_{\substack{\Lambda \in \text{WickContraction}(0) \\ \neg \text{HaveEqTime}(\Lambda)}} \text{wickTerm}(\Lambda)$$ Since the set of Wick contractions for an empty list contains only the empty contraction, which has no equal-time pairs and a Wick term of $1$, both sides of the equation evaluate to $1$.

Let $\mathcal{F}$ be a field specification and $\phi_s = [\phi_0, \phi_1, \dots, \phi_{n-1}]$ be a list of field operators. Let $\mathcal{T}$ denote the time-ordering operator and $\mathcal{N}$ denote the normal-ordering operator. For any Wick contraction $\Lambda$ on the indices of $\phi_s$, let $\text{wickTerm}(\Lambda)$ be the product $\epsilon(\Lambda, \phi_s) \cdot \text{timeContract}(\Lambda) \cdot \mathcal{N}([\phi_s]_\Lambda^{uc})$, where $\epsilon(\Lambda, \phi_s)$ is the statistical sign factor, $\text{timeContract}(\Lambda)$ is the product of pairwise time-contractions of the operators paired in $\Lambda$, and $[\phi_s]_\Lambda^{uc}$ is the sub-list of operators whose indices are not contracted by $\Lambda$. The normal-ordered version of Wick's theorem states that the time-ordered product of the normal-ordered operators is given by: $$\mathcal{T}(\mathcal{N}(\phi_0 \phi_1 \dots \phi_{n-1})) = \sum_{\substack{\Lambda \\ \neg \text{HaveEqTime}(\Lambda)}} \text{wickTerm}(\Lambda)$$ where the sum is taken over all Wick contractions $\Lambda$ that do not contain any equal-time pairs (i.e., for every pair $\{i, j\} \in \Lambda$, the operators $\phi_i$ and $\phi_j$ are not evaluated at the same time).